Graph natural log of 9x

Problem

y=ln(9*x)

Solution

1. Identify the domain of the function. Since the argument of a natural logarithm must be positive, we set 9*x>0 which means x>0

2. Determine the vertical asymptote by finding where the argument equals zero. As x approaches 0 from the right, y approaches −∞ so the vertical asymptote is x=0

3. Find the x-intercept by setting y=0 Solving 0=ln(9*x) gives e0=9*x which simplifies to 1 = 9x,o*r = \frac{1}{9}$.

4. Calculate additional points to determine the shape of the curve. For example, if x=1 y=ln(9)≈2.20 If x=e/9≈0.30 y=ln(e)=1

5. Apply the product rule for logarithms to understand the transformation. The function can be rewritten as y=ln(x)+ln(9) which is a vertical shift of the standard y=ln(x) graph upward by ln(9)≈2.20 units.

6. Sketch the graph starting near the vertical asymptote x=0 passing through (1/9,0) and increasing slowly as x increases.

Final Answer

y=ln(9*x)

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